16.21 Techniques of Structural Analysis and Design Spring 2003 Section 2-Energy and variational Principles Unit #7-Concepts of work and energy Work igure 1: Work of a force on a moving particle
16.21 Techniques of Structural Analysis and Design Spring 2003 Section 2 - Energy and Variational Principles Unit #7 - Concepts of work and energy Work Figure 1: Work of a force on a moving particle 1
Work done by a force dW=f·du=fu2=|f‖ldu‖cos(fu) dw f. du · Work done by a moment: dW=M. de= M: 0 dw Extend definition to material bodies: total work is the addition of the work done on all particles by forces distributed over the volume W by forces distributed over the surface t·udS by concentrated forces W=∑f·u(x) nother classification Work done by external forces: we will assume that external forces dont change during the motion or deformation, i. e, they are independent of the displacements. This will lead to the potential character of the external work and to the definition of the potential of the erternal force as the negative of the work done by the external forces Work done by internal forces: the internal forces do depend on the deformation In general, the work done by external forces and the work done by nternal forces don't match(we saw that part of the work changes the kinetic energy of the material)
� � � • Work done by a force: dW = f · du = fiui = �f� �du� cos(fu� ) (1) � B � B WAB = dW = f · du (2) A A • Work done by a moment: dW = M · dθ = Miθi (3) � B � B WAB = dW = M · dθ (4) A A • Extend definition to material bodies: total work is the addition of the work done on all particles: – by forces distributed over the volume: W = f · udV V – by forces distributed over the surface: W = t · udS S – by concentrated forces: n W = fi · u(xi) i=1 Another classification: • Work done by external forces: we will assume that external forces don’t change during the motion or deformation, i.e., they are independent of the displacements. This will lead to the potential character of the external work and to the definition of the potential of the external forces as the negative of the work done by the external forces. • Work done by internal forces: the internal forces do depend on the deformation. In general, the work done by external forces and the work done by internal forces don’t match (we saw that part of the work changes the kinetic energy of the material). 2
收 F=g E Figure 2: Spring loaded with a constant force Example: Consider the following spring loaded with a constant force WE= Fo, F doesn't change when u goes from 0 to O Fs(u)du, Fs: force on spring kudu WE≠W Remarks 3
Figure 2: Spring loaded with a constant force Example: Consider the following spring loaded with a constant force: WE = Fδ, F doesn’t change when u goes from 0 to δ (5) = mgδ (6) WI = � δ 0 Fs(u)du, FS : force on spring (7) = � δ 0 kudu = 1 2 kδ2 (8) ⇒ WE �= WI (9) Remarks: 3
WE=WI would imply 8= 219, which contradicts equilibrium:8 rium. How can you explain this? teached the system is not in equilib- before the final displacement d is Strain energy and strain energy density Figure 3: Strain energy density Strain energy and strain energy density(see also unit on first law of thermodynamics U=UdV 10) From first law at
˙ � � • WE = WI would imply δ = 2mg k , which contradicts equilibrium: δ = mg k , • before the final displacement δ is reached the system is not in equilibrium. How can you explain this? Strain energy and strain energy density Figure 3: Strain energy density Strain energy and strain energy density (see also unit on first law of thermodynamics): U = UdV (10) V From first law: ∂U� = σij �ij ∂t 4
au Jiideii I. not necessarily linear elastic Ciiklekde 11) Complementary strain energy and complementary strain energy density CA Figure 4: Complementary strain energy density 5
˙ ˙ � � � � � � ∂U� �ij � �� � ∂�ij = σij �ij ???? ∂U� σij = ∂�ij � �ij 0 U = σijd�ij , not necessarily linear elastic Linear case: U = �ij 0 Cijkl�kld�ij = 1 Cijkl�kl�ij = 2 1 σij �ij (11) 2 Complementary strain energy and complementary strain energy density Figure 4: Complementary strain energy density V Uc = UcdV (12) � σij 0 Uc = �ijdσij (13) 5